11,444 research outputs found
A second infinite family of Steiner triple systems without almost parallel classes
For each positive integer n, we construct a Steiner triple system of order v=2(3n)+1 with no almost parallel class; that is, with no set of v-13 disjoint triples. In fact, we construct families of (v,k,Ī»)-designs with an analogous property. The only previously known examples of Steiner triple systems of order congruent to 1 (mod 6) without almost parallel classes were the projective triple systems of order 2n - 1 with n odd, and 2 of the 11,084,874,829 Steiner triple systems of order 19
Steiner Triple Systems Without Parallel Classes
We construct Steiner triple systems without parallel classes for an infinite number of orders congruent to 3 (mod 6). The only previously known examples have order 15 or 21
Counting Steiner triple systems with classical parameters and prescribed rank
By a famous result of Doyen, Hubaut and Vandensavel \cite{DHV}, the 2-rank of
a Steiner triple system on points is at least , and equality
holds only for the classical point-line design in the projective geometry
. It follows from results of Assmus \cite{A} that, given any integer
with , there is a code containing
representatives of all isomorphism classes of STS with 2-rank at most
. Using a mixture of coding theoretic, geometric, design
theoretic and combinatorial arguments, we prove a general formula for the
number of distinct STS with 2-rank at most contained
in this code. This generalizes the only previously known cases, , proved
by Tonchev \cite{T01} in 2001, , proved by V. Zinoviev and D. Zinoviev
\cite{ZZ12} in 2012, and (V. Zinoviev and D. Zinoviev \cite{ZZ13},
\cite{ZZ13a} (2013), D. Zinoviev \cite{Z16} (2016)), while also unifying and
simplifying the proofs. This enumeration result allows us to prove lower and
upper bounds for the number of isomorphism classes of STS with 2-rank
exactly (or at most) . Finally, using our recent systematic
study of the ternary block codes of Steiner triple systems \cite{JT}, we obtain
analogous results for the ternary case, that is, for STS with 3-rank at
most (or exactly) . We note that this work provides the first
two infinite families of 2-designs for which one has non-trivial lower and
upper bounds for the number of non-isomorphic examples with a prescribed
-rank in almost the entire range of possible ranks.Comment: 27 page
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Identical twin Steiner triple systems
Two Steiner triple systems, each containing precisely one Pasch configuration which, when traded, switches one system to the other, are called twin Steiner triple systems. If the two systems are isomorphic the systems are called identical twins. Hitherto, identical twins were only known for orders 21, 27 and 33. In this paper we construct infinite families of identical twin Steiner triple systems
Uniform hypergraphs containing no grids
A hypergraph is called an rĆr grid if it is isomorphic to a pattern of r horizontal and r vertical lines, i.e.,a family of sets {A1, ..., Ar, B1, ..., Br} such that Aiā©Aj=Biā©Bj=Ļ for 1ā¤i<jā¤r and {pipe}Aiā©Bj{pipe}=1 for 1ā¤i, jā¤r. Three sets C1, C2, C3 form a triangle if they pairwise intersect in three distinct singletons, {pipe}C1ā©C2{pipe}={pipe}C2ā©C3{pipe}={pipe}C3ā©C1{pipe}=1, C1ā©C2ā C1ā©C3. A hypergraph is linear, if {pipe}Eā©F{pipe}ā¤1 holds for every pair of edges Eā F.In this paper we construct large linear r-hypergraphs which contain no grids. Moreover, a similar construction gives large linear r-hypergraphs which contain neither grids nor triangles. For rā„. 4 our constructions are almost optimal. These investigations are motivated by coding theory: we get new bounds for optimal superimposed codes and designs. Ā© 2013 Elsevier Ltd
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